Appendix A. the ratio of Curvature Radiation (CR) to ICS power
The energy loss of a particle through CR process is:
where is the Lorentz factor of the particles, e is the electric charge of the particles, c the light of speed, is the radius of curvature of the magnetic field. The curvature radius for a dipole magnetic field is , , is the radius of the light cylinder, is a constant for a dipole magnetics field line for last opening field lines the is:
The energy loss of a particle through ICS is:
and is the total cross-section of ICS, is the photon number density near the surface, is the energy of the outgoing photons, near the surface, and in the estimate below. Near the surface of the neutron star, the photon number density of the low frequency wave can be written as:
where E is the electric field in the gap. For the inner gap sparking, one expects that the low frequency wave with electric field has the value of E. In the limit (h is the thickness of the gap, is the radius of the gap), we have (RS75):
Where B is the magnetic field near the surface of the neutron star. Substitute into Eq. (A3), we have:
Thus the ratio of the energy loss of these two processes near the surface of the neuron star is:
If we use (RS75), the ratio can be increase about times, but is still very small. This means that near the surface of the neutron star the efficiency of CR compare which the ICS is very low. Above the surface and
At a distance r, from the equations above, we have:
where (see below). Even if the typical radiation height is at in Eq. (A11), is still small. This means that the efficiency of the ICS process is higher than that of the CR process, but as the estimate bellow, the incoherent radiation of ICS process is inadequate in explaining pulsar radiation either.
Appendix B. average number densities of the outflow
The energy flux carried by relativistic positrons into the magnetosphere above two polar gaps are (RS75):
where the current form in the magnetosphere is taken as (Sutherland, 1979):
Here . The current associated with the corotating plasma of charge density is (Goldreich and Julian 1969):
If the magnetospere is charge separated then the number density of the particles of charge is
The second current corresponds to streaming of charges along the magnetic field lines: and k must be constant along a given field line. This current only exist on the open field lines. The exact division between the open and closed field lines cannot be determined precisely. We may use the vacuum dipole field geometry to locate approximately the division on the neutron star and use the Eq. (B4).
If the potential in the inner gap has the maximum value (RS75):
Thus, . This means that the current flow from the gaps have enough braking torque on the spinning star to lose all the rotational energy (Sutherland 1979). This is reasonable, because . Even if in the charge separated magnetosphere, n0 can be much larger than the GJ density . Especially, in the place near the stellar surface. Near stellar surface, in the balance between the gravitational force and kinetic energy of the particles lets a thin atmosphere existence. We will estimate the luminosity of the ICS process using Eq. (B4).
Appendix C. luminosity in incoherent ICS processes
The incoherent Luminosity by the ICS process near the surface of the neutron star is:
where is a constant, is characteristic time over which the particle radiates at some desirable frequency.
From Eq. (6) for constant and , we have: . As , we can get from Eq. (11) and we can get from (generally , since ), thus, . As an estimation, we take , , so second.
Substitute Eqs. (A4)(B4) and (C2) to Eq. (C1), we can get the luminosity as follows:
where is in order of 1, is the Thomson cross section.
Here the index "R " presents the parameters in the electron rest frame. Here we indicate by 1 (1') the linear polarization of the incoming (outgoing) photon parallel to the plane built up by the magnetic field and the incoming (outgoing) photon, and by 2 (2') the linear polarization orthogonal to this plane. , the cyclotron frequency. In our case , so , and
where is the angle between the direction of the incoming photon and the magnetic field. As ,
At , using Eq. (A10) and (C7), if the radiation region is at and , the luminosity in Eq. (C3) will be down 9 magnitudes. The luminosity observed for radio pulsars can be written as (Sutherland,1979):
The range of is from to , which means that incoherent ICS radiation is inadequate in explaining pulsar radiation.
© European Southern Observatory (ESO) 1998
Online publication: April 15, 1998