**Investigation of the Magnetosphere
of Ganymede with Galileo's Energetic Particle Detector**

*Ph.D. dissertation by Shawn M. Stone, University of Kansas,
1999.*

*Copyright 1999 by Shawn M. Stone. Used with
permission.*

**3.4 Inertial Reference Frames and Coordinate Systems**

An inertial reference frame is defined as a system that is neither rotating nor accelerating relative to a certain reference point. The differential equations of motion are then represented by constant velocity terms. This provides a standard frame from which physical quantities such as position, velocity, and acceleration can be measured. The method of describing such a system in space science begins with the definition of the Z axis as the spin vector of the Earth (geographic north pole). The X axis is defined as the point where the equator of the Earth crosses the ecliptic plane of the solar system at the vernal equinox of a specific Julian date (epoch). The Y axis completes the right handed system. The stationary point that this system is referenced to is given by the star background. The current catalog of stars used is the FK4 star map, which consists of 1500 stars, and their position pins down the orientation of the inertial reference frame [U.S. Naval Observatory, 1983; Kovalevsky et al., 1989]. This geometry is shown in Figure 3.8. The Galileo spacecraft uses the position of these stars to calculate its orientation in J1950.

Figure 3.8 The
inertial reference frame defined by the position of
the vernal equinox γ at a
certain epoch (J1950 or J2000). The X axis is
defined by the point where the ecliptic plane
crosses the equator at γ at
the relevant epoch. The Z axis is defined by the
spin axis of the Earth (North Pole). The Y axis
completes the orthogonal set of the right handed
inertial reference frame. For epoch J1950, 1500
stars have been cataloged from the barycenter of the
solar system; this is called the FK4 catalog. A
spacecraft, such as Galileo, finds its orientation
from the positions of these stars [Kovalevsky et
al., 1989]. |

The orientation of any object in the solar system can be
calculated from the knowledge of the right ascension α and
declination δ angles referenced in
an equatorial system of the respective epoch. The epochs in
use today are Julian date 1950 (J1950) and Julian date 2000
(J2000). For example, the orientation of the north pole of a
body is specified by its right ascension α_{o} and
declination δ_{o}, while the location of the prime
meridian from this point is specified by the angle W [Davies
et al., 1980]. Table 3.5 shows some of these values for
epoch J1950 for the Sun, Earth, and Jupiter.

**Table 3.5** The
standard equatorial coordinates for the north pole of the
respective body in J1950. T=interval in Julian ephemeris
centuries from standard epoch and d-interval in ephemeris
days from the standard epoch [Davies, 1980].

Body |
Angles |

Sun | α_{0}=286.0°δ _{0}=63.8°W=240.9° + 14.184° d |

Earth | α_{0}=0.0°- 0.640° Tδ _{0}=90.0°-0.556° TW=99.87°+360.98° d |

Jupiter | α_{0}=286.00°-0.008° Tδ _{0}=64.5°+.003° TW _{III}=80.6°+870.53° d |

*3.4.1 Jovian Centered Coordinate Systems*

The Jovian systems that are pertinent to this work are
the Jupiter system III(1965) and Jupiter Magnetic system III
coordinate systems. From the rate of modulation of the
decametric radiation, the spin period of Jupiter has been
calculated to be 9h 55m 29.37s [Riddle and Warwick, 1976;
Berge and Gulkis, 1976]. The Jupiter system III(1965)
coordinate system (Figure 3.9a) rotates with the planet at
its spin rate. The z axis is the spin axis, the x axis is
the system III longitude λ_{III}(1965)=0, and y
completes the orthogonal set. This system is a left-handed
system with longitude increasing from East to West according
to an observer at Earth. Jupiter Magnetic system III (Figure
3.9b) is based on the orientation of Jupiter's magnetic
field relative to the JSIII(1965) system whose angles are
based on a dipole whose magnetic axis is tilted by 9.6°
relative to the JSIII(1965) Z axis and λ_{III}(1965)=
201.7° east longitude. The Z axis is the magnetic dipole
axis, the X axis lies within the plane of the JSIII XZ
plane, and the Y axis completes the orthogonal set.

Figure 3.9 (A)
Jupiter system III(1965) coordinates. The Z axis is
defined by the spin axis of Jupiter. The X axis is
defined by the system III longitude λ_{III}=0
(prime meridian). The Y axis completes the
orthogonal left handed system. (B) Jupiter Magnetic
system III. This system is defined by the
orientation of the magnetic dipole axis in JSIII.
The orientation of the dipole axis is given as θ=9.6°
and λ_{III}=201.7° [Acuna and Ness,1976].
Rotation of the dipole axis into the Z axis while
maintaining the X axis of the JSMIII in the plane of
the JSIII XZ plane fixes this coordinate. |

*3.4.2 Ganymede Centered Coordinate Systems*

The coordinate system of Ganymede can be defined in a number of ways. The longitude on Ganymede, or any other satellite for that matter, is measured from its prime meridian, which is the sub-Jovian meridian. This meridian is fixed on the Jovian satellites because the same face of each always points toward Jupiter. For Ganymede System I (Figure 3.10a), the X axis is in the direction of corotation (or orbital direction), the Y axis points toward Jupiter along the sub-Jovian meridian, and the Z axis is the spin axis of Ganymede, which can be assumed to be nearly aligned with the spin axis of Jupiter. Ganymede System II (Figure 3.10b) is rotated by 90° about the Z axis so that the X axis points toward Jupiter and the Y axis is in the anti-corotation direction.

Figure 3.10 (A)
Ganymede system I coordinates. The Y axis points
radially towards Jupiter. The X axis points in the
direction of corotation. The Z axis completes the
right handed orthogonal set. (B) Ganymede system II
coordinates. This system is rotated by 90° about the
GSI Z axis. The X axis points toward Jupiter, the Y
axis in the anti-corotation direction, and Z
completes the right handed orthogonal set. |

*3.4.3** **Magnetic Field
and Spacecraft Coordinates*

The definition of magnetic field coordinates begins with the magnetic field vector at a point in the Jovian magnetosphere. The magnetic field vector is at that point defined as the Z axis. The X axis lies in the plane defined by this Z axis and the radius vector from Jupiter to the point in question. The Y axis completes the orthogonal set of this right handed coordinate system (Figure 3.11a). The pitch angles of the particles are as defined in Equation [2.6], and the phase angle is measured relative to the primed X axis according to Equation [2.7]. Spacecraft coordinates are defined relative to the spin axis of the Galileo spacecraft which is the Z axis. The Y axis is the vector normal to the step platform of the EPD instrument. The X axis completes the orthogonal set (Figure 3.11b).

Figure 3.11 (A)
Magnetic field coordinates. The Z axis is defined as
the magnetic field vector direction. The X axis is
fixed by the radius vector from Galileo to Jupiter
which lies in the XZ plane. The Y axis completes the
orthogonal set. (B) Spacecraft coordinates. The Z
axis is the spin axis of the spacecraft. The Y axis
is defined as the rotor axis for the stepping motor.
The X axis completes this set. |

Next: 3.5 Coordinate Transformations and SPICE Kernels

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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.