3. The thermal photon distribution
To compute Eq. (4) and (13) we need an expression for the differential photon distribution of the thermal radiation from the hot polar cap of the neutron star. For the sake of simplicity we regard the symmetrical case of an electron sitting centered and above the polar cap at a height d. Further, let us assume a homogeneous and isotropic emission from a hot thermal cap. We consider two source geometries:
with a the neutron star radius and the angular velocity of the pulsar.
If is the distance between the electron and the considered emission point at the thermal cap and is the angle between s and d, we can write for the flux density at the position of the electron and coming from the direction
where is a differential (ring-)surface element of the thermal cap and denotes the opening angle of this ring-element seen from the center of the neutron star.
The maximum value allowed for is given either by the border of the hot thermal cap or (for small values of d) when s equals the tangent line from the electron to the neutron star (i.e. when the border of the cap is behind the horizon). The first case is expressed by
To compute expression (17) has to be always smaller than (20) and (21).
We note that Eq. (17) for the differential photon distribution is physically equivalent to the one of Dermer (1990).
© European Southern Observatory (ESO) 2000
Online publication: May 3, 2000