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Subsections

Definitions

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

Nomenclature

The Cluster PEACE experiment is made up of two sensor heads called LEEA and HEEA (Low and High Energy Electrostatic Analysers), mounted on opposite sides of each of the four Cluster spacecraft (Johnstone et al (1997)). Generally LEEA scans the lower energy range and HEEA the upper range, together covering electrons with $0.7$ eV $< E < 26$ keV. Over the duration of a spin ($\sim 4$ s), LEEA and HEEA cover 4$\pi$ steradians of velocity space.

The moment sums can be thought of as combinations of the low energy moments (the B (BOTTOM) moments, $E \sim 10$ eV), the high energy moments (the T (TOP) moments, $E \gtrsim 2$ keV), and the moments from where the energy scan of LEEA and HEEA overlap (which we call L1L2 and H1H2 respectively) as sketched in Figure 3.2. These separate pieces are telemetered to the ground, where we can construct moment sums covering the entire energy range by summing B, L1L2/H1H2 and T moments. The `measured' moments (subscript `m') are the raw sum of, for example B+overlap+T moments, which have not been corrected.

Figure 3.2: Schematic representation of the PEACE sensor coverage. The two sensor heads HEEA and LEEA are split into an energy range which is divided into 88 bins. The sensor coverages can overlap by some amount (left), leaving a low energy portion (BOTTOM) and a high energy portion (TOP). The sensors can be made to overlap completely (right) leaving no BOTTOM or TOP moments.

Image energy_levels


next up previous contents
Next: The algorithm Up: qtmc_manual Previous: Spacecraft moments: introduction   Contents
Steve Schwartz 2005-03-26