**
Ulysses HISCALE Data Analysis Handbook**

## Appendix 10. Effect of Backscattered Electrons on the Geometric Factors of the LEMS30 Telescope (Hong MS Thesis)

where ΔA'_{ij} is
the effective area of ΔA_{i} (see
Figure A10-7) looking into the solid angle of ΔΩ_{ij}. ΔΩ_{ij} is
the solid angle of the jth escaped electron trajectory; n is
the total number of finite areas ΔA_{i}, and npas
is the total number of escaped trajectories for ΔA_{i} (see Section
A10-6 for detailed derivations). To
calculate the Ω_{ij} one can trace an
electron starting from the open aperture with a particular
energy and see if it can reach the area ΔA_{i} of
the detector. We also can do it time-reversed, starting at
the center of ΔA_{i} and trace its
trajectory to see if it escapes the open aperture. In this
project we use the second procedure. Since we would compare
the results of the effect of backscattering electrons on
geometric factors of LEMS 30 to Buckley's non-scattering
results, it is important to use his model of calculation.
The procedure is outlined as follows:

1) The telescope has a complex geometric shape. In order to determine the fate of a particle's trajectory (check if the particle should continue its trajectory) we need a simulation of the geometry of the telescope. Buckley modeled the whole LEMS 30 subsystem with plane polygons (see Figures A10-4 - A10-5). A complete listing of the coordinates of each vertex and coefficients of each plane can be found in Section A9.5.

Figure A10-4 View angle θ=0°, Φ=90°

Figure A10-5 View angle θ=-20°, Φ=80°

2) The approximation of the magnetic field generated by the magnets is also needed. Buckley adapted Shodhan's model (Shodhan, S., Masters Thesis, Univ. of Kansas, 1988) to do the calculation for the LEMS 30 telescope.

3) Divide the detector sensor into 21 small areas of ΔA_{i} (see
Figure A10-6). Choose a ΔA_{i}.
(For the coordinates of the center and area of each ΔA_{i} see
section 9.6.)

Figure A10-6 The
detector is divided into 21 small ΔA_{i}.

4) Start from the center of
the small area ΔA_{i}, choose initial θ_{0} and Φ_{0} with
a particular energy for a small time interval (Δt_{j}). During
the time interval the particle moves from (x_{j}, y_{j}, z_{j}) to
(x_{j+1}, y_{j+1}, z_{j+1}) under the influence of the magnetic
field.

5) Determine whether the line segment formed by
(x_{j}, y_{j}, z_{j}) and
(x_{j+1}, y_{j+1}, z_{j+1}) has an intersection with any
plane of the polygons of 1.

- If there is no intersection then calculate the
trajectory of the electron for the time interval
(Δt
_{j+1}); repeat step 5. - If there is an intersection between the line segment
and a polygon then check to determine if the polygon is
the open aperture polygon.
- If it is open aperture, then terminate the trajectory because the electron escapes the open aperture.
- If it is not, then determine if the trajectory
has any earlier impact with the telescope.
- If there is no earlier impact, use a particular backscattering model (see Section A10.2) to reschedule the trajectory after the impact.
- If there was an earlier impact, then terminate the trajectory and the electron fails to be detected.

6) Repeat step 4 with change of Δθ and ΔΦ until it exhausts all possible combinations of θ and Φ.

7) Pick another ΔA_{i} and repeat
step 4. Notice that because of the symmetry of the detector
we only need to pick ΔA_{i} in a half
of the detector. The results of the other half should be the
same (see Figure A10-7).

Figure A10-7 The number
of escapes at each ΔA_{i} for energy
of 50 keV (Radzimski's η is used).

Figure A10-7 shows the number
of escapes for energy of 50 keV for the electrons starting
from the center of each ΔA_{i} when
Radzimski's backscattering coefficients are used. The
results include elastic specular single backscattering
electrons. More results can be found in Section
A10.9 for both Radzimski's and Neubert's η and
other energies.

8) One can use the results for the particular energy of the electrons to calculate the geometric factor by using the following formula:

where n is total number of ΔA_{i}'s
on the detector, ΔA_{i} is the area
of the i-th element on the detector, Δθ_{j} is
the interval at which the polar angle is chosen to scan
through the whole solid angles, θ_{j} and Φ_{j} are
the polar and azimuthal angles at which the j-th electron
starts from the center of the area which escapes the open
aperture, and npas is the number of trajectories which reach
the open aperture. The 25º that appears in the argument of
cosine is needed because the detector B surface tilts in
this coordinate system. See Section
A10.6 for more details.

One can repeat all the above procedures to obtain the geometric factors for different energies. Shown in Figure A10-8 are the geometric factors vs. energies for including elastic specular single backscattering when Radsimski's η is used.

For details of the programming and codes see Section A10.8 and Section A10.11.

Figure A10-8 The geometric factor including specular backscattering; Radzimski's η is used.

Next: A10.4 Chapter 4 - Results and Discussion

Return to the Table of Contents for Hong's MS Thesis

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Updated 1/2/19, Cameron Crane

## QUICK FACTS

**Manufacturer:**ESA provided the Ulysses spacecraft, NASA provided the power supply, and various others provided its instruments.

**Mission End Date:**June 30, 2009

**Destination:**The inner heliosphere of the sun away from the ecliptic plane

**Orbit:**Elliptical orbit transversing the polar regions of the sun outside of the ecliptic plane