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Astron. Astrophys. 362, 383-394 (2000)

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3. Specific cross-sections

3.1. Hard X-ray (HXR) moments

Energetic electrons moving through a plasma will experience close (relative to the collisions dominating their energy loss and scattering) collisions with protons and ions that result in the emission of HXR bremsstrahlung radiation. This is described by the well known Bethe-Heitler cross-section [FORMULA], which has mathematical form:

[EQUATION]

where [FORMULA] represents the emitted photon energy, [FORMULA]m2 keV, and [FORMULA], the Heaviside step function, is 1 for [FORMULA] and zero otherwise. Note that L and [FORMULA] are defined to be zero when the electron energy E is less than the photon energy [FORMULA]. This allows us to keep the limits of the energy integrals in the moments as 0 and [FORMULA].

Consider the single particle expectation [FORMULA], where q is some function of [FORMULA] that is of interest. [FORMULA] is the HXR emission rate due to a single particle, per unit photon energy. The quantity [FORMULA] therefore represents an averaging of q over the emission of a single particle. More generally, we can define the average over the whole electron distribution f, using the the expectation operator [FORMULA] defined by (5) or (7) with [FORMULA] given by (8):

[EQUATION]

This equation can be used to relate an observable (left hand side), such as the centroid of the HXR emission at a particular photon energy [FORMULA], to terms on the right hand side that will involve moments of the injected distribution h, such as the mean injected pitch angle or the mean injected position. Referring to equation (1), the quantity [FORMULA] is the rate of HXR photon emission at energy [FORMULA], per unit [FORMULA], at time t, from the whole electron distribution described by f.

3.2. Heating moments

Heating moments are very similar in form to the HXR moments, and are in fact simpler because there is no introduction of a further variable, such as the photon energy [FORMULA]. The cross-section is given by

[EQUATION]

where [FORMULA] and e is the electronic charge (in e.s.u.) and [FORMULA] is the Coulomb logarithm, which we assume to be constant. Despite the [FORMULA] dependence and the lower limit of the energy integrals in the moments being [FORMULA], the integrals do not diverge because of the presence of nv in the integrands.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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