# Wind/SWE "New mode" Electron Pitch-angle Analysis

## Overview

Effectively, in this mode of the electron instrument, there are 13
energy-channels (electron speeds) making 48 observations (instrument
counts for each particular velocity) over 8 sectors (\~45° wide) and 6
elevations (\~9° apart) at each energy. From these 624 observations
(accumulated over 3 s/c spins, or \~9s) 13 pitch-angle distributions and
one "spin-averaged" energy distribution are populated during every \~12s
interval.

![](media/image1.jpeg){width="6.0in" height="4.5in"}

**In the following discussion:**

-   The observed energies are represented as E~0~, E~1~, ..., E~12~.
    Where E~0~, E~1~, ..., E~10~ are observed *once* in each sector;
    E~11~, E~12~ are observed *twice*, then averaged. This achieves more
    statistically meaningful observations in energy-ranges where fewer
    counts are observed, and accounts for the mapping from the 15
    energy-channels in the current operational mode to the 13 channels
    discussed here.

-   i = 0, 1, ..., 5 indexes *detector*.

-   j = 0, 1, ..., 7 indexes *sector*.

-   k = 0, 1, ..., 12 indexes *energy*.

-   **B**, **V** vectors are given in GSE coordinates, with **B** being
    the temporally "nearest" 3s MFI observation to any given SWE
    observations of electron **V**.

-   α~i,j,k~ = cos^-1^{**V**~i,j,k~ • **B**(t) **/** \[\|**V**~i,j,k~\|
    \|**B**(t)\|\] } gives the pitch-angle for each observation, with
    c~i,j,k~ giving the corresponding count-value.

![](media/image2.png){width="5.625in" height="7.5in"}

**For E = E~k~, the 30 pitch-angle bins (6° wide) of the k^th^ (of 13)
pitch-angle distribution (f~k~(α)) are populated as follows...**

Typically a conversion from counts to phase-space densities requires
both energy and a "geometric factor" for each contributing detector.
However, acknowledging that the following process unavoidably
intermingles counts from different detectors, and that the geometric
information from our instrument specifications gives a single conversion
factor (call it cf~k~) for each energy (E~k~), averaged count-values are
simply scaled by this factor to yield corresponding phase-space
densities:

f~k~(α) = \[Σc~i,j,k~ / n\] cf~k~, α = 3°, 9°, ..., 177°.

The summation is carried out over i,j s.t. α~i,j,k~ lies within 3° of
the specified center-angle α (e.g. from 0° to 6° for α = 3°, 6°-12° for
α = 9°, ..., 174°-180° for α = 177°), and n is the number of detectors
(observations) contributing counts to the sum.

One additional value is generated for each energy:

\<f\>~k~ = \[Σc~i,j,k~ / N\] cf~k~

where the summation is carried out over *all* i,j and N is the number of
detectors contributing counts to this "grand total" sum.

**This completes the information needed to generate our [electron
pitch-angle data product]{.underline} for this mode (very similar to
that of previous modes):**

f~0~(α), f~1~(α), ..., f~12~(α) for α = 3°, 9°, ..., 177° (30 6° bins)
form the pitch-angle distributions, and \[\<f\>~0~, \<f\>~1~, ...,
\<f\>~12~\] forms the "spin-averaged" energy (speed) distribution,
accumulated over each \~9s interval (available about every 12s).

**For reference:**

-   The 6 GSE *particle-velocity direction* elevation angles used in our
    analysis are:

> -26.55°, -17.10°, -7.34°, 7.63°, 17.10°, and 26.53° (above the
> eccliptic).

-   The 13 electron energies (E~k~ k = 0, 1, ..., 12) are:

> 19.34, 38.68, 58.03, 77.37, 96.71, 116.1, 193.4, 290.1, 425.5, 580.3,
> 773.7, 1006., and 1238. eV.

-   The 13 corresponding electron speeds are:

> 2.61, 3.69, 4.52, 5.22, 5.83, 6.39, 8.25, 10.1, 12.2, 14.3, 16.5,
> 18.8, and 20.9 x 10^8^ cm/s.

## Derived Products

From the electron pitch-angle distribution product described above,
there are derived two additional data products: *the 'strahl' product*
(useful both as a reduced summary of the pitch-angle product, and as an
indication of whether the strahl phenomenon is being observed), and *the
'pitch-average' product* (another approach to reducing the pitch-angle
product, this time by averaging over regimes of interest). Basic
derivation and contents of these products is discussed below. Note that,
as with the pitch-angle product, these products are very similar to
those of previous modes.

**[Strahl Product]{.underline}: First, recall the 30 pitch-angle bins
(6° wide) of the k^th^ (of 13) pitch-angle distribution (f~k~(α) for E =
E~k~) populated above...**

![](media/image3.png){width="6.0in" height="4.5in"}

Next, a smoothing operation is applied to the "measured" values. Then we
perform the following analysis on both the "**B** side" (the 15 bins
associated with forward-streaming electrons) and the "anti-**B** side"
(the other 15 bins associated with backward-streaming electrons). The
maximum *measured* value is noted (call it M~B~, and M~aB~,
respectively); giving a characteristic intensity for each side. After
this, the maximum *smoothed* value is noted, as well as the
corresponding half-maximum, and a search is conducted starting from this
position. We seek the first bin, in each direction, whose value falls at
or below the half-maximum value. We call the pitch-positions of the
smoothed maxima α~M-B~, and α~M-aB~, respectively. The separation of the
*half*-maximum bins we call W~B~, and W~aB~, respectively; as they give
characteristic widths.

When the strahl phenomenon is being observed, these parameters describe
the strahl "beams" being observed. In this case, M~\*~ (\* = B or aB)
provides a measure (in units of phase-space density) of beam intensity,
α~M-\*~ provides a representative pitch-angle for each beam, and W~\*~
provides a measure of (angular) beam-width. One typical use of this
product (which is entirely consistent with previous modes of the
electron instrument) is as an indicator of whether the strahl phenomenon
is, in fact, being observed. When the phenomenon *is* being observed, we
expect to see relatively intense beams with relatively small deviations
from the 0° and 180° pitch-angle positions and relatively small widths.
An investigator interested in this phenomenon could focus attention (on
the underlying high-resolution data) only where these criteria were met.
On the other hand, this product is always useful as a reduced, summary
representation of the pitch-distribution product; as it tends to capture
(in a small amount of data) the sort of salient features apparent in our
visualizations of the unreduced pitch product.

13 values (from pitch-angle distributions at each energy) of M~B~,
M~aB~, α~M-B~, α~M-aB~, W~B~ and, W~aB~ (available about every 12s) form
this data product.

**[Pitch-average Product]{.underline}: Derived, again, from the 30
pitch-angle bins (6° wide) of the k^th^ (of 13) pitch-angle distribution
(f~k~(α) for E = E~k~) populated above...**

![](media/image4.jpeg){width="6.0in" height="4.5in"}

The figure for this section suggests the final approach for reducing our
pitch-angle data: *averaging over regimes of interest on the **unit
sphere***. This approach begins by identifying 28 annular "strips" (and
two "polar caps") of solid angle by the relation α~i~ \< α \< α~f~,
where the α~i~ and α~f~ values are the familiar pitch-angle bin
boundaries from above. Next, we establish "weights" for our averaging
process by calculating:

Ω~m~ = ~αi,m~ ∫ ^αf,m^ \[2π sin α\] dα = 2π \[cos α~i,m~ -- cos α~f,m~\]

for the m^th^ pitch-angle bin. Upon finally noting that the 0° \< α \<
90°, 60° \< α \< 120°, and 90° \< α \< 180° regions all have solid angle
2π str---the entire sphere having solid angle 4π str; we may proceed
with averaging. The results (k^th^ pitch-distribution) are:

f~para,k~ = 1/2π Σ~m=0,14~ \[f~k~(α~m~) Ω~m~\], ("parallel" bins, where
0° \< α \< 90°)

f~perp,k~ = 1/2π Σ~m=10,19~ \[f~k~(α~m~) Ω~m~\], ("perpendicular" bins,
where 60° \< α \<120°)

f~anti,k~ = 1/2π Σ~m=15,29~ \[f~k~(α~m~) Ω~m~\], ("anti-parallel" bins,
where 90° \< α \< 180°)

f~omni,k~ = 1/4π Σ~m=0,29~ \[f~k~(α~m~) Ω~m~\], (*all* bins, where 0° \<
α \<180°)

13 values of f~para,k~ (f~para,0~, f~para,1~, ..., f~para,12~), along
with the corresponding 13 values of f~perp,k~, f~anti,k~, and f~omni,k~
(available about every 12s) form this data product.

## Data-product Files for this Mode

Each data-product described above is stored as both a "flat binary" file
and a CDF file for each day of processed data. For reference, a typical
day yields 6000-7000 energy "spectra" (about 1 every 12s, with data
accumulated over 9s; accounting for "null spectra", data gaps, etc.).
Each non-null spectrum yields the set of values described above. Noting
that the naming conventions are mostly historical in nature (with the
exceptions of 'h' for "high resolution" and 'm' for "modified", as in
derived, in the CDF files), the files for each *yyyymmdd* date are
named:

  --------------------- ----------------------- -------------------------------
  **Product**           **Flat-binary           **ISTP-compliant CDF filename**
                        filename**              

  ***Pitch*-angle       yyyymmdd_v06.pit        wi_h3_swe_yyyymmdd_v01.cdf
  distributions**                               

  **Pitch-distrib.      yyyymmdd_v06.pitavg     wi_m0_swe_yyyymmdd_v01.cdf
  *averages***                                  

  ***Strahl*            yyyymmdd_v08.str        wi_m1_swe_yyyymmdd_v01.cdf
  beam-descriptions**                           
  --------------------- ----------------------- -------------------------------

[Notes]{.underline}: The first date of processed new-mode data is
***yyyymmdd*=20020816**. While accessing the data from our flat-binary
files requires the specialized software used internally by the SWE
electron science team, the CDF files are self-describing and a variety
of software tools exist for accessing their data.