## 3. The thermal photon distributionTo 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 -
(a) The thermal photons come from the polar cap bordered by the open field lines (see below). This corresponds to the case where the surface is heated by the particle current. -
(b) The photons originate from a larger area around the magnetic pole, e.g. . This corresponds to the case of internal heating.
The polar cap radius (case (a) above) in the standard model of an aligned rotator is given by: with For the photon spectrum we use a black body model. Therefore, the energy flux per energy element and solid angle element at any surface point within the thermal cap can be written as If is the distance between the
electron and the considered emission point at the thermal cap and
is the angle between 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. As we consider a rotational symmetric situation we can identify the angle with the angle in Sect. 2 and therefore can write for the angular photon density with . The maximum value allowed for is
given either by the border of the hot thermal cap or (for small values
of with the radius of the hot thermal cap (for case (a) equal to given by Eq. (14), for case (b) set to ) and the second 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 |