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Adding high energy moments

In practice, we restrict the inversion of (4.8-4.10) to moment sums which exclude the TOP moments of PEACE. This improves the numerical accuracy and speed of the algorithm, since high energy moments do not require correcting as the presence of a finite spacecraft potential has neglibible effect. These moments (described by the TOP moments in the case of PEACE) are incorporated into the overall sum in the following way.

Given the meaured onboard moments for $v < v_u$, that is, $(N_m, (N{\bf V})_m, {\bf P}_m)$, we use our correcting algorithm to yield the characterising features $(N_c, V_c, T_c)$ of the Maxwellian distribution $f(v)$ (equation 3.6). The direction of the velocity is identical to the measured velocity direction by virtue of the scalar approximation. We then perform three 1d integrations on the system $g_{1,2,3}$ over the truncated range $0 \leq v \leq v_u$ with a null spacecraft potential to restore the corrected moments into the truncated-corrected form $(N_{tc}, (NV)_{tc}, {\rm Tr}({\bf P}_{tc}))$:

\begin{displaymath}
N_{tc} = g_1\left[N_c, V_c, T_c, \Phi_{sc}=0\right]_{v_L=0}^{v_U=v_u}
\end{displaymath} (4.15)


\begin{displaymath}
(NV)_{tc} = g_2\left[N_c, V_c, T_c, \Phi_{sc}=0\right]_{v_L=0}^{v_U=v_u}
\end{displaymath} (4.16)


\begin{displaymath}
{\rm Tr}({\bf P}_{tc}) =g_3\left[N_c, V_c, T_c, \Phi_{sc}=0\right]_{v_L=0}^{v_U=v_u}
\end{displaymath} (4.17)

where
\begin{displaymath}
{\rm Tr}({\bf P}_{tc}) = 3p_{tc} + 2m_e(NV)_{tc}V_c - m_eN_{tc}V_c^2
\end{displaymath} (4.18)

The direction of $(NV)_{tc}$ is restored by the measured velocity direction, and the corrected stress tensor is reconstructed into a velocity aligned frame thus:
\begin{displaymath}
{\bf P}_{tc} = p_{tc}{\bf I} + m_eN_{tc}(2V_{tc}V_c - V_c^2){\bf\hat{V}}_m{\bf\hat{V}}_m
\end{displaymath} (4.19)

From here the TOP moments, assumed to reflect the integrals of $f(v)$ from $v_u \rightarrow \infty$, can be added to the truncated-corrected moments yield the final corrected moments:
\begin{displaymath}
N_f = N_{tc} + N_T
\end{displaymath} (4.20)


\begin{displaymath}
(N{\bf V})_f = (N{\bf V})_{tc} + (N{\bf V})_T
\end{displaymath} (4.21)


\begin{displaymath}
{\bf P}_f = {\bf P}_{tc} + {\bf P}_T
\end{displaymath} (4.22)


next up previous contents
Next: Temperature anisotropy estimation Up: The algorithm Previous: The normalised non-linear system   Contents
Steve Schwartz 2005-03-26