Foundations of Space Dynamics. Ashish Tewari

serve as useful references for both navigation and orientation of the vehicles.

Figure 1.3 The equinoctial sidereal frame , the ecliptic synodic frame , and Earth centred celestial meridian.

       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 in Fig. 1.3. Another axis of the sidereal frame can be taken to be normal to either the ecliptic or the equatorial plane (axis in Fig. 1.3), and the third axis can be chosen to be perpendicular to the first two (axis in Fig. 1.3).

      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 (). Such a correction gives the sidereal day as the following:

      (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 in Fig. 1.3, rotates very slowly against a background of distant stars. Hence the vernal equinox (and the equinoctial sidereal reference frame) can be approximated to be the fixed references for most space flight applications. However, for a long flight time of several years' duration, the calculations must be brought to a common reference at a specific time (called an epoch1) by applying the necessary corrections, which take into account the slow movement of the vernal equinox towards the west. The equinox is given for various epochs by the International Earth Rotation and Reference Systems Service (IERS) in terms of the longitude of the equinox measured from a celestial meridian (see Fig. 1.3). The inclination of Earth's spin axis from the normal to the ecliptic is called the obliquity of the ecliptic (Fig. 1.3), and also varies with time due to the nutation caused by the luni‐solar attraction. (The precession and nutation, discussed in detail in Chapter 11, cause Earth's spin axis to rotate with time due to the luni‐solar attraction.) The value of the obliquity of the ecliptic in the current epoch is measured by IERS to be about 26'21”. The period of nutation of Earth's spin axis is about 41000 yr., which is considerably longer than the period of its precession. The precession and nutation are explained in Chapter 11 when considering the rotation of a rigid body (such as Earth).

      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 tropical year is not the true period of revolution of Earth in its orbit around the sun. The true period of revolution is the sidereal year, which is measured by timing the passage of Earth against the background of distant stars, and equals 365.25636 mean solar days. Thus a tropical year is shorter than the actual year by 20 hr., 40 min., and 42.24 s.

       1.2.2 Celestial Frame

For a motion taking place inside the solar system, any two stars (except the sun) appear to be fixed for the duration of the motion. Hence, a reference frame constructed out of three mutually perpendicular axes, each of which are pointing towards different distant stars, would appear to be fixed in space, and can serve as a sidereal reference frame. A reference frame fixed relative to distant stars is termed a celestial reference frame. For example, the rate of rotation of Earth about its own axis can be measured by an observer standing astride the North Pole by timing the rate