*Astron. Astrophys. 362, 383-394 (2000)*
## 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
, which has mathematical form:
where represents the emitted
photon energy,
m^{2} keV, and
, the Heaviside step function, is 1
for and zero otherwise. Note that
*L* and are defined to be zero
when the electron energy *E* is less than the photon energy
. This allows us to keep the limits
of the energy integrals in the moments as 0 and
.
Consider the single particle expectation
, where *q* is some function of
that is of interest.
is the HXR emission rate due to a
single particle, per unit photon energy. The quantity
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
defined by (5) or (7) with given by
(8):
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
, 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
is the rate of HXR photon emission
at energy , per unit
, 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 .
The cross-section is given by
where and *e* is the
electronic charge (in e.s.u.) and is
the Coulomb logarithm, which we assume to be constant. Despite the
dependence and the lower limit of
the energy integrals in the moments being
, the integrals do not diverge
because of the presence of *nv* in the integrands.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 19100
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