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Subsections

The algorithm

Choice of frame

Our choice of calculation frame simplifies greatly the numerical technique. We transform the spacecraft measurement frame via a rotation $\mathcal{R}$ which aligns the $z$ axis with the measured velocity ${\bf V}_m$ such that ${\bf V}_m = (0, 0, \vert{\bf V}_m\vert)$. The scalar quantity $N_m$ remains the same, as does the trace of the pressure tensor. The transformed quantities can be directly inferred from the measured values, so the details of the rotation matrix do not need to be known. As only the magnitude of the velocity is needed, we can use the measured direction of the velocity to recover the corrected velocity vector from the speed derived in our calculation.

The algorithm converges to a solution by improving on a set of initial guesses. The values of initial guess we use were derived from a series of tests in which, given a set of Maxwellian parameters, we simulated measured moments given a range of $\Phi_{sc}$ and energy cut-offs. We ran our algorithm on these inputs to recover the initial underlying Maxwellian values. The initial guesses were then chosen to be the average values for which the alorithm converged successfully for a range of parameter space which represents typical plasma environments encountered by spacecraft.

The normalised non-linear system

We normalise the unknown moments in terms of the measured ones as:
\begin{displaymath}
V_e^{\prime} = \frac{\vert{\bf V}_e\vert}{V_{m}}
\end{displaymath} (4.1)


\begin{displaymath}
V_{Te}^\prime = \frac{V_{Te}}{V_{Tm}}
\end{displaymath} (4.2)


\begin{displaymath}
\eta = \frac{1}{\sqrt{\pi}}\frac{N_e/N_m}{V_e^\prime V_{Te}^\prime }.
\end{displaymath} (4.3)

Where $V_{Te}$ is the thermal speed, from which the temperature can be inferred:
\begin{displaymath}
T_e = \frac{m_eV_{Te}^2}{2k}
\end{displaymath} (4.4)

The inputs to the solver are the normalised quantities
\begin{displaymath}
\zeta_{sc} = \frac{V_{m}}{V_{Tm}}
\end{displaymath} (4.5)


\begin{displaymath}
\epsilon_{sc} = \frac{\mathcal{E}}{V_{Tm}^2}
\end{displaymath} (4.6)


\begin{displaymath}V_{L,U} = \frac{v_{L,U}}{V_{Tm}}\end{displaymath} (4.7)

For the inputs $v_L=0$, $v_U = \infty$, $\mathcal{E} = 0$, the exact solution yields unity for equations 4.5 & 4.6 and $1/\sqrt{\pi}$ for equation 4.7.

The set of non-linear equations which must be inverted are:
\begin{displaymath}g_1(N_e,V_e,T_e) - N_m = 0 \end{displaymath} (4.8)


\begin{displaymath}g_2(N_e,V_e,T_e) - N_mV_{m} = 0 \end{displaymath} (4.9)


\begin{displaymath}g_3(N_e,V_e,T_e) - 3N_mkT_m - m_eN_mV_m^2 = 0 \end{displaymath} (4.10)

That is, the measured moments are functions of the real moments. We recast to a normalised system such that we wish to find the triplet ($\eta$, $V_e^\prime$, $V_{Te}^\prime$), given the normalised inputs ($\zeta_{sc}$, $\epsilon_{sc}$, $V_{L,U}$). The equations to be solved are
\begin{displaymath}
1-\frac{\eta}{\zeta_{sc}}\int_{V_L}^{V_U} \sqrt{V^2-\epsilon_{sc}}\left(E^- - E^+\right){\rm d}V = 0
\end{displaymath} (4.11)


$\displaystyle \zeta_{sc}^2-\eta\int_{V_L}^{V_U} (V^2-\epsilon_{sc}) \left[E^- + E^+
-\frac{V_{Te}^{\prime 2}(E^-E^+)}{2VV_e^\prime\zeta_{sc}}\right]{\rm d}V = 0$     (4.12)


$\displaystyle \frac{3}{2} + \zeta_{sc}^2 - \frac{\eta}{\zeta_{sc}}\int_{V_L}^{V_U} (V^2-\epsilon_{sc}) ^ {3/2}
\times(E^- - E^+){\rm d}V = 0$     (4.13)

where
\begin{displaymath}
E^{\pm}=\exp{\left[-\left(\frac{V\pm V_e^\prime\zeta_{sc}}{V_{Te}^\prime}\right)^2\right]}
\end{displaymath} (4.14)

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)

Temperature anisotropy estimation

The simplification of the method proposed in GS and implemented here lies in the reduction to a 1d integration, made possible through the use of the scalar approximation and an isotropic distribution. However we can attempt to estimate the anisotropy - i.e. the random velocities parallel and perpendicular to the magnetic field - and interpret the result as two temperature components $T_{\parallel}$ and $T_{\perp}$. Here we describe the estimation.

The $ij^{{\rm th}}$ element of the measured pressure tensor ${\bf p}_m$ can be expressed

\begin{displaymath}p_{m_{ij}} = P_{m_{ij}} - m_e(NV)_{mi}V_{mj}\end{displaymath} (4.23)

which can be decomposed in terms of the unit magnetic field vector ${\bf b}$ in the case of a gyrotropic distribution:
\begin{displaymath}p_{m_{ij}} = p_{m\parallel}b_ib_j + p_{m\perp}(\delta_{ij}-b_ib_j).\end{displaymath} (4.24)

The trace of ${\bf p}_m$ is then
\begin{displaymath}{\rm Tr}({\bf p}_m) = p_{m\parallel} + 2p_{m\perp}.\end{displaymath} (4.25)

Finally the measured anisotropy $A_m$ is found as
\begin{displaymath}
A_m = \frac{p_{m\perp}}{p_{m\parallel}} = \frac{{\rm Tr}({\bf p}_m) - p_{m\parallel}}{2p_{m\parallel}}.
\end{displaymath} (4.26)

We make the assumption that this measured anisotropy is an approximation to the underlying anisotropy and so we can partition the corrected temperature $T_f$ according to:
\begin{displaymath}
3T_f = T_{f\parallel} + 2T_{f\perp}
\end{displaymath} (4.27)

with
\begin{displaymath}
\frac{T_{f\perp}}{T_{f\parallel}} \equiv A = A_m
\end{displaymath} (4.28)

As a check we rotate ${\bf p}_m$ into a field-aligned frame such that:

\begin{displaymath}
\left[\begin{array}{ccc}
p_{m00} & p_{m01} & p_{m02}\\
p_{m...
...{m\perp}_1} & 0 \\
0 & 0 & p_{{m\perp}_2}
\end{array} \right]
\end{displaymath}

In practice the off-diagonal terms are not identically zero and we use the ratio of these to the diagonal terms as an error check. Additionally $p_{m\perp 1} \neq p_{m\perp 2}$ provides a second check.

Estimation parameters

We calculate three estimation parameters which describe how much the measured on-board moments were over- or under-estimated:
\begin{displaymath}
r_N = \frac{N_m - N_f}{N_f}
\end{displaymath} (4.29)


\begin{displaymath}
r_V = \frac{V_{m} - \vert{\bf V}_f\vert}{\vert{\bf V}_f\vert}
\end{displaymath} (4.30)


\begin{displaymath}
r_T = \frac{{\rm Tr}({\bf p}_m)/3N_mk - T_f}{T_f}
\end{displaymath} (4.31)


\begin{displaymath}
I_e = \sqrt{r_N^2 + r_V^2 + r_T^2}.
\end{displaymath} (4.32)

These parameters trace the effect that various environments have on the measured moments. A more general estimation parameter $I_e$ is yielded by the Pythagorean combination of the three parameters above. Generally $I_e < 1$ (that is, $<60$% correction on all moments). As mentioned above, the amplitude of the over- or under-estimation is determined by environment, however their general behaviour is also a function of spacecraft potential (which itself is somewhat influenced by the characteristics of the ambient plasma). GS describe how the estimation parameters vary for a range of potentials in three plasma environments: the solar wind, magnetosphere and magnetosheath. Those authors conclude that the moments are affected worst in the solar wind, where $I_e$ can exceed 60%. In the other regions the moments are affected to a lesser extent, but in general still require correcting. GS and Salem et al (2001) also describe the existence of a critical potential, $\Phi_{{\rm crit}}$, for which the ratio $r_N$ is zero, and no correction is required (no such critical point exists in general for the temperature or velocity). In this circumstance the energy range truncation (resulting in an under-sampled distribution) is compensated by the potential broadening caused by $\Phi_{{\rm crit}}$ such that, despite truncation and the presence of a potential, the density integration over $f(v)$ returns the correct value.


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