## The Galileo Energetic Particles Detector

## Galileo EPD Handbook

### Chapter 1. Instrument Summary

**Charged Particle Response of Magnetic Deflection System
for Galileo Jupiter Orbiter**** ****(draft)***
(continued)*

**Magnetic Field Model**

The magnetic field is due to a uniform magnet lying in the x-y plane with its center at the origin; its integral form [7] is given by

where σ_{M} is the surface density
of the magnetic pole strength. M is the magnetization, which
is constant, and its direction is the same as the normal of
the magnet. The factor M/c is converted to the normalization
constant B_{0} which was used to fit experimental
values of the field.

We obtained this three-dimensional, analytic expression
for B(x,y,z) by evaluating the integral [8].
Its components are given in equations (5), (6), and (7).
From these expressions, we constructed a complete analytic
expression for the field provided by both separate magnets.
The experimental measurements of magnetic field strength are
only available in the middle plane (z=0.0) where, because of
the symmetry of the sensor, B_{x}=B_{y}=0 should vanish [9]. The
two magnets are placed in a pattern: one at the left tilted
counter-clockwise by an angle, the other tilted at the right
clockwise by an angle of the same degree. For this
arrangement, the expressions of both magnets require a
coordinate rotation as well as a translation of the
coordinate system. These transformations are given in
equations (8), (9), and (10).

where a is the half-width of the magnet, l the
half-length of the magnet, and B_{0} the
normalization constant of the magnetization.

The rotation of the coordinates for the left magnet is given by

the rotation of the coordinates for the right magnet by

and the translation of the coordinates for both magnets by

where y'_{1} and z'_{1} are for the
left magnet, y'_{2} and z'_{2} are for
the right magnet, d is the separation between the left and
right magnets, ψ is the angle by
which the magnets are tilted, and x_{1}, y_{1}, and z_{1} are
the distances shifted from the origin of the coordinates.

In addition, in order to simulate the actual measurements of the field, the two magnets are further divided into smaller pieces. For each of the smaller magnets, the rotation and translation were carried out. The magnetization values of all small magnets are obtained by normalizing the maximum field value from the experimental data. Figure 3 provides the distribution of the magnetizations of these magnets.

Figure 3. Comparison between
measurements and results of the model. Each curve
corresponds to a different specific value of y
(left) and x (right). |

The superposition of these small magnets (both left and right) results in the total field strength and gives a good agreement with the experimental data inside the sensor (see Figure 3). The magnetic field lines are computed and projected on both the y-z and the x-z plane as given in Figure 4. These field lines converge to the fringes of the magnets. The contour plots in Figure 5 demonstrate the expected variation of the field and show the divergence, and the curl of the magnetic field vanishes inside the sensor. The results show that the edges of the magnets are highly magnetized and the field is stronger where the two magnets are closer. The numerical results of the magnetic field of the magnetic field and the parameters used in the simulation are provided elsewhere [10].

Next: Methods of Calculation

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Updated 8/23/19, Cameron Crane

## QUICK FACTS

**Manufacturer:**The Galileo Spacecraft was manufactured by the Jet Propulsion Laboratory, Messerschmitt-Bölkow-Blohm, General Electric, and the Hughes Aircraft Company.

**Mission Duration:**Galileo was planned to have a mission duration of around 8 years, but was kept in operation for 13 years, 11 months, and 3 days, until it was destroyed in a controlled impact with Jupiter on September 21, 2003.

**Destination:**Galileo's destination was Jupiter and its moons, which it orbitted for 7 years, 9 months, and 13 days.