coupled, such as when the rigid body experiences an appreciable gravity‐gradient torque during its orbit. Furthermore, when designing an attitude control system for a spacecraft, it is necessary to account for its orbital motion. Therefore, while elements of orbital mechanics and attitude dynamics can be grasped separately, their practical application involves a combined approach.
1.2 Reference Frames and Time Scales
Space flight requires a definite background of objects to measure distances, as well as to orient the spacecraft in specific directions. Since fixed objects are hard to come by in practice, navigation and attitude determination are non‐trivial problems in space flight. Such a problem does not exist for the motion taking place on, or very close to, a solid surface, where ground‐fixed objects can serve as useful references for both navigation and orientation of the vehicles.
Figure 1.3 The equinoctial sidereal frame
1.2.1 Sidereal Frame
Three mutually perpendicular straight lines joining distant objects constitute a reference frame. Generally, distant objects in the universe are moving with respect to one another; hence the straight lines joining them would rotate, as well as either stretch out or contract with time. Suppose one can find two objects which are fixed relative to each other. Then a straight line joining them would be fixed in length, and a vector pointing from one object to the other would always have a constant direction. A reference frame consisting of axes which have fixed directions is said to be a sidereal frame. There are certain directions which can be used to orient a sidereal frame. For example, the orbital plane of Earth around the sun, called the ecliptic, intersects Earth's equatorial plane along a straight line called the line of nodes. The nodes are the two specific points where this line intersects Earth's orbit, as shown in Fig. 1.3. One of the two nodes is an ascending node, where the apparent motion of the sun as seen from Earth (called the apparent Sun) occurs from the south to the north of the equator. This happens at the vernal equinox, occurring every year around March 21. The descending node of the apparent sun is at the autumnal equinox, which takes place around September 22. Since the vernal equinox points in a specific direction from the centre of Earth, it can be used to orient one of the axes of the sidereal frame, as the axis
The rate of rotation of Earth on its own axis (normal to the equatorial plane) is from the west to the east, and can be measured in a sidereal reference frame oriented with the vernal equinox direction. This rate is called the sidereal rotation rate, and would be the true rotation rate of Earth if the vernal equinox were a constant direction. A sidereal day is the period of rotation of Earth measured from the vernal equinox. If the sun is used for timing the rotational rate of Earth, the period from noon to noon is a mean solar day (m.s.d.) of 24‐hour duration. However, the mean solar day is not the true rotational rate of Earth because of Earth's orbit around the sun, which also takes place from the west to the east. To calculate the sidereal day from the mean solar day, a correction must be applied by adding the average rate at which Earth orbits the sun. The tropical year is the period of Earth's orbit around the sun measured from one vernal equinox to the next, and equals 365.242 mean solar days. This implies that the mean apparent sun is slightly less than one degree per day (
(1.3)
or 23 hr., 56 min., 4.0904 s.
Unfortunately, the vernal equinox is not a constant direction because of the slow precession of Earth's axis (thus the equatorial plane) caused by the gravitational influence of the sun and the moon (called the luni‐solar attraction). When a spinning rigid body, such as Earth, is acted upon by an external torque, such as due to the gravity of the sun and the moon, its spin axis undergoes a complex rotation called “precession” and “nutation”, which will be explained in detail in Chapter 11. This rotation of the equatorial plane causes the two equinoxes to shift towards the west, and is thus called the precession of the equinoxes. The period of the precession is about 25772 yr., which implies that the sidereal day differs only slightly from the true rotational period of Earth. It also means that an equinoctial sidereal reference frame, such as the frame
Apart from the precession and the nutation of Earth's spin axis, there is also a precession of the ecliptic caused by the gravitational attraction of the other planets. This is a much smaller variation in the equinoxes (about 100 times smaller than that caused by luni‐solar attraction).
Since the vernal equinox moves slightly westward every year, the
