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Moments of the velocity distribution function

The $n^{\rm {th}}$ moment of a distribution $f({\bf v})$ is defined as:

\begin{displaymath}{\bf M}_n = \int f({\bf v}){\bf v}^n{\rm d}^3v\end{displaymath} (3.1)

Certain combinations of the moments from an electron velocity distribution have a familiar physical interpretation:
\begin{displaymath}
N = M_0
\end{displaymath} (3.2)


\begin{displaymath}
N{\bf V} = {\bf M}_1
\end{displaymath} (3.3)


\begin{displaymath}
{\bf P} = m_e{\bf M_2}
\end{displaymath} (3.4)


\begin{displaymath}
{\bf p} = {\bf P} - m_e{\bf M_1}{\bf V}
\end{displaymath} (3.5)

where $N$ is the density, ${\bf V}$ the bulk velocity vector, ${\bf P}$ the stress tensor and ${\bf p}$ the pressure tensor. We assume that in free space (i.e. far away from the spacecraft where there are no effects from the potential), the distribution has a Maxwellian shape at temperature $T$, and drifts with some velocity ${\bf V}$:
\begin{displaymath}f({\bf v}) = N\left(\frac{m_e}{2 \pi k T}\right)^{3/2}\exp\left(-\frac{m_e}{2kT}\vert{\bf v} - {\bf V}\vert^2\right) \end{displaymath} (3.6)

Integration over all solid angles and energies yields the number density $N$. It is easy to see therefore that in general for a finite integration range $v_l < \vert{\bf v}\vert < v_u$, as used in real detectors, $N$ will be underestimated. Note that the amount the on-board moments are under- or over-estimated is also a function of spacecraft potential (see GS), such that for large potentials the density is over-estimated.

Near the spacecraft the energy conservation of an electron can be expressed:
\begin{displaymath}v_m^2 = v^2 - \mathcal{E}\end{displaymath} (3.7)

where $v$ is the electron velocity in free space (hereafter a subscript `m' denotes a parameter as measured by the spacecraft and `c' denotes output from the solver. `sc' is a value inherent to the spacecraft, such as potential) and $\mathcal{E}$ corresponds to the free space energy of an electron which arrives at the detector with zero energy:
\begin{displaymath}\mathcal{E} = -\frac{2e\Phi_{sc}}{m_e}.\end{displaymath} (3.8)

$\mathcal{E}$ is negative for most of the plasma environments in space, since typically the spacecraft potential $\Phi_{sc} > 0$ due to the escape of photo-electrons. Liouville's theorem tells us that the distribution function is constant along a phase space trajectory
\begin{displaymath}f(v_m, \theta_m, \phi_m) = f(v, \theta, \phi).\end{displaymath} (3.9)

GS make the scalar approximation: namely that only the magnitude of the velocity is affected by the potential (i.e. $\theta_m=\theta$ and $\phi_m=\phi$). Under this assumption the angular dependence in the moment integrations can be performed analytically, thus reducing the problem to one dimension. By changing the integration element $v_m{\rm d}v_m = v{\rm d}v$ from equation 3.7, the measured moments can be written in terms of $v$ and $\mathcal{E}$. The integration limits are related to the detector cut-offs $v_{l,u}$ by
\begin{displaymath}v_{L,U} = \sqrt{v_{l,u}^2+\mathcal{E}}.\end{displaymath} (3.10)

In the case of $v_l^2 + \mathcal{E} < 0$ where the potential reaches a value greater than the lower energy cut-off, we set $v_L=0$. Figure 3.1 is a schematic representation of the measured part of the distribution compared to the corrected part and the underlying (assumed) Maxwellian distribution.
Figure 3.1: Schematic showing the effect of a positive spacecraft potential on an idealised drifting Maxwellian electron distribution (dotted curve). On-board calculated moments are integrals under the shifted segments (solid grey) which can be writtten, with the help of Liouville's Theorem, in terms of the idealised distribution. Inverting this then yields the Maxwellian parameters.
Image shifted_dist


next up previous contents
Next: Nomenclature Up: Definitions Previous: Definitions   Contents
Steve Schwartz 2005-03-26