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Astron. Astrophys. 357, 301-307 (2000)

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3. The thermal photon distribution

To compute Eq. (4) and (13) we need an expression for the differential photon distribution [FORMULA] 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:

  • (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. [FORMULA]. 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:

[EQUATION]

with a the neutron star radius and [FORMULA] the angular velocity of the pulsar.

For the photon spectrum we use a black body model. Therefore, the energy flux per energy element [FORMULA] and solid angle element [FORMULA] at any surface point within the thermal cap can be written as

[EQUATION]

If [FORMULA] is the distance between the electron and the considered emission point at the thermal cap and [FORMULA] 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 [FORMULA]

[EQUATION]

where [FORMULA] is a differential (ring-)surface element of the thermal cap and [FORMULA] 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 [FORMULA] with the angle [FORMULA] in Sect. 2 and therefore can write for the angular photon density [FORMULA]

[EQUATION]

with [FORMULA] following from Eq. (15)

[EQUATION]

and the geometric term

[EQUATION]

with [FORMULA].

The maximum value allowed for [FORMULA] 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

[EQUATION]

with [FORMULA] the radius of the hot thermal cap (for case (a) equal to [FORMULA] given by Eq. (14), for case (b) set to [FORMULA]) and the second case is expressed by

[EQUATION]

To compute expression (17) [FORMULA] has to be always smaller than [FORMULA] (20) and [FORMULA] (21).

We note that Eq. (17) for the differential photon distribution is physically equivalent to the one of Dermer (1990).

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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